1. Field of the Invention
This invention relates to complete helical cone-beam scanning and non-redundant data acquisition for three-dimensional imaging of arbitrarily long objects.
2. Description of Related Art
A two-dimensional detector 11 and a point-shaped ray source (e g X-ray) S are assumed to move synchronously around the object in a helical trajectory as shown in FIG. 1. In a medical tomograph the helical source movement is achieved by translating the patient through the rotating source-detector gantry with constant speed. Two-dimensional projections are acquired (detected) at arbitrarily short intervals along the trajectory. The detector 16 consists of a large number of sensors (detector elements) which are evenly spaced and placed in a plane or, as in FIG. 1, on the surface of the helix cylinder 12. Although the rotation axis 14 is normally horizontal in medical tomographs, rather than vertical as in FIG. 1, we will adopt the following convention. In the sequel, vertical means a direction parallel to the rotation axis 14 (the z-axis) in FIG. 1, while horizontal means a direction parallel to the xy-plane 15.
A projection consists primarily of intensity measures for the incoming rays to a detector element. The logarithms of these primary data represent the sum of the attenuation along the rays, i.e. line integrals over the three-dimensional attenuation function f we want to retrieve. But to be able to reconstruct f from its projections in a correct way, all points in the object have to be fully exposed and the projection data utilized in a balanced way. Thus, if back-projection is used for the reconstruction, projections from all projection angles must be available and brought in with the correct weight to obtain what is called exact reconstruction. Also, the projection data have to be filtered correctly to compensate for the inherent low-pass filtering in the projection-back-projection procedure.
In the literature several mathematically exact methods have been proposed for reconstruction from cone-beam projections. In most cases these methods demand that the object is of finite extension, i.e. restricted in size, so that its total projection never falls outside the available detector. Unfortunately, this requirement is not realistic in most cases of computer tomography, e.g. when one is to reconstruct a full body, or long objects in general. Traditionally, 1D-detector arrays are made large (wide) enough to cover the object across its maximum width. However, for several reasons, it is out of question to extend these 1D-detectors to 2D-detectors, which cover the patient from head-to-toe. Instead, in the foreseeable future, available 2D-detectors will be used to cover and record projections of a section of a long object.
Today, three-dimensional volume data are reconstructed slice-by-slice. The patient is translated slowly (typically 2 mm/sec) while the X-ray source and a one-dimensional detector array are synchronously and continuously rotated at speeds of around 1 r/sec. Relative to a patient which is not moving, the source and detector are then performing a helical movement with very low pitch, say, 2 mm. The reconstruction employs a modified versions of traditional 2D-reconstruction methods for circular scanning of a single slice. However, with the given numbers, it takes approximately 100 sec to fetch data for a 200 mm long section of the body. During this time, due to breathing and other body functions, the body is not fully at rest which blurs the reconstructed object. A second drawback is that the anode of the X-ray tube is subjected to severe strain and extreme temperatures during longer exposure times.
In a 1D-detector system the major part of the generated photons are collimated away without being utilized, while a 2D-detector system is able to utilize a substantial part of these otherwise wasted photons. Hence, by using a 2D-detector with, say, n parallel 1D-detectors in the above example, the velocity can be increased to 2n mm/sec and the scanning time reduced to 100/n sec. Alternatively, speed can be traded for strain on the X-ray source so that, for instance, if the photon flow is halved, the velocity is more moderately increased to n mm/sec and the scanning time reduced to 200/n sec. However, in any case it is no longer possible to perform the reconstruction using conventional 2D-methods since the projection rays are no longer, not even approximately, in the same plane during one turn of the source trajectory.
Circular Source Trajectory
A well-known method for inexact reconstruction from cone-beam projections taken along a circular path was proposed in [Feld84]. The 2D-detector is placed on a planar surface and extended horizontally to cover the width of the object. The width of the object and its distance to the source defines the maximum fan-angle .gamma..sub.max of the source-detector system. In the vertical direction the planar detector is limited by two horizontal lines. Along the vertical axis where these lines are closest to the source we find the maximum cone-angle. The image reconstruction consists of the following steps taken for each detector recording. All corrections of geometrical and radiometric nature, including the ever necessary logarithm computation have been left out here for the sake of brevity.
1 Pre-weighting of the recorded detector data with a factor that is proportional to the cosine of the angle between the central ray and the ray that originated the detected value. PA1 2. Filtering with traditional ramp-filtering techniques along each horizontal detector row. PA1 3. Back-projection along the original ray in which process the filtered detector value is multiplied with a so called magnification factor which depends on the distance between the ray source and the object point to receive a contribution from the ray. PA1 3a. During the back-projection, for a certain projection angle, among all possible source positions which illuminate an object point, contributions are accepted only from the position which is closest to the actual point in the z-direction. PA1 1. Pre-weighting of the recorded detector data with a factor that is proportional to the cosine of the angle between the central ray in the projection and the ray that originated the detected value. PA1 2. Re-binning to complementary projections. PA1 3. Filtering of the original as well as the complementary projections with traditional ramp filtering techniques along each horizontal detector row. Because of the non-planar detector this filter is slightly different from the filter employed in [Feld84]. PA1 4. For each projection angle, filtered projection data are back-projected along the rays. The values are multiplied with magnification factors which depend on the distance between the point and ray source. All such values, from original as well as complementary cone-beam source positions, are averaged into one single contribution that is accumulated to the object point. PA1 1. Rebinning to oblique parallel projections. PA1 2. Pre-weighting of the recorded detector data with a factor that is proportional to the cosine of the angle between the ray that originated the detected value and the central ray. PA1 3. Reconstruction of one horizontal slice from generalized projections. The latter can be seen as the result of imaginary projection rays running within the horizontal slice. PA1 3.1 Computation of Fourier domain contributions to this slice for one generalized projection at every projection angle. PA1 3.1.1 Computation of Fourier domain contributions for each "detector" position in one generalized projection. PA1 3.1.1.1. Multiplication of projection data (from all source positions that send oblique rays through the slice in this position) with a pre-computed set of weights, which are Fourier series components derived from an adopted interpolation function. PA1 3.1.1.2. Summation of the contributions for each Fourier component to obtain a single set, a truncated Fourier transform for each ray in this detector position of this generalized parallel projection. PA1 3.1.2 Computation of the Fourier transform (FFT) along the projection for all these truncated Fourier components to obtain a kind of truncated 2D Fourier transform contribution for each generalized parallel projection. PA1 3.1.3. Multiplication of the Fourier transform of this generalized projection with a ramp filter. PA1 3.2. Merging of filtered data from all projection angles in the 2D Fourier space of the horizontal slice and resampling with a space-invariant interpolation filter. PA1 3.3. Application of an inverse 2D Fourier transform (FFT). PA1 3.4. Compensating for imperfect interpolation in the Fourier domain by post-weighting the result with the inverse interpolation function, in accordance with the well known gridding technique. PA1 1. From each 2D-projection, partial contributions to the derivative of Radon transform values are computed by means of line integration along a multitude of lines in the planar detector. This requires that we select a specific object point to be the origin of a 3D coordinate system. PA1 2. When the scanning is complete, that is when the helical trajectory has covered the intended target region of the object (ROI), all these partial contributions are sorted and coplanar partial contributions are summed. PA1 3. The result is resampled into a regular grid in the Radon transform space of the ROI of the object function. PA1 4. Filtering with a derivative filter. PA1 5. 3D back-projection which takes place as two consecutive 2D back-projection steps.
This method gives perfect results for image slices in, or close to the mid-section of the object. For slices which have been subjected to more oblique rays at higher cone angles, the image quality deteriorates.
Helical Source Trajectory. Non-exact Methods
Extensions of [Feld84] to helical source paths were first proposed by [Wang93]. Here, the planar 2D-detector is given a vertical extension large enough to ascertain that every point is exposed to the source at least once for every projection angle during a full 360 degree source rotation. The effect of this requirement is that for any given projection angle an object point will be exposed by the source from various numbers of source positions; at least one but often many more, depending on the given fan-angle, cone-angle and detector size. This has to be taken into account during the back-projection. Hence, [Feld84] is employed in [Wang93] but augmented with the following rule.
A way to achieve a more efficient and balanced exposure of the object points was proposed in [Scha96]. Here, the detector is located (wrapped) onto the surface of the source cylinder 41, which in FIG. 2 is seen to be centered in S. The radius of this cylinder equals the source-detector distance, which is different from the radius R of the helix cylinder 12. The helix cylinder is coaxial with the object cylinder in FIG. 1 which is defined by the maximum object width r. In [Scha96] the detector is limited in the vertical direction by two horizontal circles (cross-sections) of the source cylinder 41. However, it is not quite clear what the minimum or optimal height is to be recommended for the detector. In the horizontal direction the detector is limited by two vertical lines, set to let the detector cover the object cylinder. In the following we may use FIG. 1 to clarify some prior art such as this.
The main novelty in [Scha96] is the introduction of complementary projections. These are projection data captured at the source cylinder 41, but sorted and resampled (rebinned) with respect to where the rays from the source are reaching the helix cylinder 12. Assume for the moment that the source is moving along the helix 16 while the object and the helix cylinder is fixed. All rays from various source positions on the same turn, reaching a fixed point on the helix cylinder, is said to be a complementary fan-beam and the projection data for this fan-beam is created during the rebinning. The set of such not quite horizontal and not quite planar fan-beams, with rays fanning out from points on a vertical line on the helix cylinder, constitute a complementary projection. These are employed in the following reconstruction procedure.
We notice that the detector arrangement in [Scha96] does not secure a perfectly balanced exposure of the object points. During the back-projection event, for each rotation angle, there is a similar situation as in [Wang93] where the object points are exposed from one or several source positions. The difference is that in [Scha96] all these projection data from both original and complementary projections are utilized and averaged together during the backprojection.
[Scha97] proposes another reconstruction technique that is claimed to be more computationally efficient. The detector system is identical to the one in [Scha96] with two horizontal truncating circles on the surface of the source cylinder 41. The reconstruction consists the following steps.
The first rebinning step is best understood if the source, the detector, and the object 17 is pictured as seen from above. From there, the cone-beams will be seen as fan-beams. The rebinning in step 1 above is equivalent to sorting projection data into sets where data from this point of view are produced not by fan-beams but from parallel beams. The term oblique parallel projections stems from the fact that the rays are parallel when seen from above, but in general non-parallel and oblique to the horizontal plane. To understand the following steps it is now recommendable to imagine a planar, virtual detector 122 as in FIG. 3 placed vertically on the rotation axis in the middle of the object. There are several source positions which produce the rays for this projection. Since the real detector on 41 is truncated horizontally and the source positions are located along the helix 16, the effective area of this virtual detector does not have a left-right symmetric shape. The upper and lower boundaries 131 and 132 are curved and tilted as shown in FIG. 4. This is a difference to the perfectly rectangular shape of the corresponding virtual detector 72 in FIGS. 10A and 10B for the present invention. The net effect is that in [Scha97] a varying number of source positions, generate fan-beams which penetrate a given slice under various oblique angles. All of these contribute to the result in the above ingenious but rather complicated computation steps 3.1.1.1 and 3.1.1.2. In the present invention we will find one and only one such fan-beam.
Helical Source Trajectory. Exact Methods
An exact method for reconstruction of a limited sector, a Region Of Interest (ROI) of a long object was proposed in [Tam95] and [Eber95]. The helical scanning covers the full vertical extension of the ROI but has to be complemented with two circular scans at top and bottom, respectively. The detector is placed on a planar surface, just as in [Feld83] and [Wang93] but the detector window is limited to the area between two consecutive turns of the source trajectory 16 as in FIG. 1. The upper and lower truncating lines on the detector plane are therefore neither horizontal, straight, nor left-right symmetric. The arguments for the specific extension of the detector stem from a well-known completeness condition for Radon planes which carries over to the following reconstruction technique. In essential aspects this method is an outgrowth from [Gra87].
Generally speaking, this reconstruction is more complicated and costly than the previous ones. Also, the rhythm of the reconstruction procedure is affected by the chosen size of the ROI. It does not feature an even flow of identical procedures repeatedly taking place for every new projection regardless of the length of the object. The two extra circular scans are highly unwanted since they break the smooth and continuous translation-rotation motion of the helical part. However, the method is optimal in one respect. For a given pitch of the helix it utilizes a minimum sized detector. Citation of a reference herein, or throughout this specification, is not to construed as an admission that such reference is prior art to the Applicant's invention of the invention subsequently claimed.